Soon after the result of [1]}, economists have realized the deficiencies of second-best mechanisms and the search for alternatives has begun almost immediately. For example, Chatterjee and Samuelson [2]} offer the following k-double auction: the seller submits a bid \(r_s\) , the buyer submits \(r_b\) . If \(r_b<r_s\)  the sell price is set to be \(k\cdot r_b+(1-k)r_s\) . For the case of \(k\in (0,1)\) , Satterthwaite and Williams [3]} and Leininger, Linhart and Radner [4]} show that the auction (which is not incentive compatible) may exhibit a continuum of equilibria. The efficiency of these equilibria vary significantly: some of these mechanisms might guarantee no efficiency at all, whereas others might guarantee the efficiency of second-best mechanisms.
